Timeline for Understand Riemannian cross-derivative on product manifolds
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2023 at 17:21 | vote | accept | Jason Li | ||
| Sep 26, 2023 at 6:09 | history | edited | Willie Wong | CC BY-SA 4.0 |
added 16 characters in body
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| Sep 26, 2023 at 6:07 | comment | added | Willie Wong | Another way to think about items 1 and 2: your manifold inherits a product Riemannian structure and hence a Levi-Civita connection, with respect to which you can define the Hessian of $f$. The musical operations allow you to consider the operation $H^\sharp_f: T(M\times N) \to T(M\times N)$, which is self-adjoint since the Levi-Civita connection is torsion free. The product structure gives the canonical decomposition $T_{x,y}(M\times N) = T_xM \times T_y N$ and your cross derivative is just the off-diagonal parts of $H^\sharp_f$. | |
| Sep 26, 2023 at 6:01 | history | answered | Willie Wong | CC BY-SA 4.0 |